The Slater and Sub-k-Domination Number of a Graph with Applications to Domination and k-Domination

In this paper we introduce and study a new graph invariant derived from the degree sequence of a graph G, called the sub-k-domination number and denoted sub_k(G). This invariant serves as a generalization of the Slater number; in particular, we show that sub_k(G) is a computationally efficient sharp lower bound on the k-domination number of G, and improves on several known lower bounds. We also characterize the sub-k-domination numbers of several families of graphs, provide structural results on sub-k-domination, and explore properties of graphs which are sub_k(G)-critical with respect to addition and deletion of vertices and edges.